SOLUTION: Please show work Consider the function f(x)=x^5-x^3 Determine the behaviors as x→∞ and as x→-∞ Find all zeros. What is the multiplicity of roots (z

Algebra ->  Functions -> SOLUTION: Please show work Consider the function f(x)=x^5-x^3 Determine the behaviors as x→∞ and as x→-∞ Find all zeros. What is the multiplicity of roots (z      Log On


   

Question 1020275: Please show work
Consider the function
f(x)=x^5-x^3
Determine the behaviors as x→∞ and as x→-∞
Find all zeros. What is the multiplicity of roots (zeros)?
When will the function cross the x-axis?
When will it cross the y-axis?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Let's do some modification.
f%28x%29=x%5E5-x%5E3
f%28x%29=x%5E3%28x%5E2-1%29
So as x get large in the positive direction, the right hand side is positive and the left hand term is positive so,
lim%28x-%3Einfinity%2Cf%28x%29=infinity%29
SImilarly, as x get large in the negative direction, the right hand side is still positive and the left hand term turns negative so,
lim%28x-%3Einfinity%2Cf%28x%29=-infinity%29
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Factor even more,
f%28x%29=x%5E3%28x%5E2-1%29
f%28x%29=x%5E3%28x%2B1%29%28x-1%29
So there are zeros at x=-1,x=0,x=1.
The root at x=0 has multiplicity of 3.
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Odd multiplicity on the zero root means it crosses three times.
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When x=0,
y=0
(0,0)
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